What is the least amount of money I can get away with signing 15 draftees with?

Whatever your round 11-25 guys are asking for. Set your draft settings to very conservative for rounds 11-25 so you don't get any above slot guys on your list. Those 15 guys should sign for somewhere between $300k - $400k in total

What happens if you don't sign 15 guys?

You don't have to sign any guys if you don't want to. Nothing happens. I assumed you were asking so you could transfer funds from your prospect budget to your player payroll budget. You need to sign at least 15 guys to transfer money beneath the $6M prospect budget threshold.

Don't be the guy that doesn't sign any of his draft picks. If too many teams do this in a league it fvcks up the minor league player pool and teams across the entire league have a hard time filling their minor systems.

What if you transfer the money, then fail to sign the guys?

Oh, you mean to transfer FROM prospect budget, transferring to doesn't matter?

Right, transfer from. Everyone is required to set aside at least $6M to sign their draft picks. If you sign 15, the game allows you to transfer the remaining (well really only $4M, assuming you only put in $6M to begin with, since transfers have to be in $2M increments).